There are several puzzle types I enjoyed elsewhere, that I'd really like to see here.

Trigons: I'll just link to an Example PDF. It has all the instructions on it already. Math Maze: Again, I can't remember if this was what it was called. Dell had a puzzle where you must draw a line through a grid of square tiles connecting the entrance and exit. At the start of each row and column is a number. You must place dots in the squares such that each row and column contains exactly the listed amount, and the dots must connect to form a continuous, non-overlapping line from the start to the finish. Hidato: Wikipedia link. Just create a path of increasing consecutive numbers from the starting tile (1) to the ending tile (N). Consecutive numbers can be orthogonally or diagonally adjacent. Fillomino: Wikipedia link. Not unlike Nurikabe, but still considerably different. You fill in borders between squares instead of blacking out squares to make N-size sections. Every number N must belong to a section of N-size; BUT, not every section needs to contain exactly one number.